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Astronomy Object of the Month

The cosmological model with a second-order perturbations

The standard cosmological model is given by the Friedmann-Lemaitre-Robertson-Walker solution. It explains many aspects of the observed Universe. However, the success of this model is accompanied by some mysteries and problems to be solved. The model predicts the existence of unknown, hypothetical forms of matter like dark matter and dark energy.

cosmological model

Illustration: Exemplary isodensity surfaces of the considered cosmological model. Credit: The Authors.


Moreover, there appears a discrepancy between the local observations of the Lemaitre-Hubble constant value and its value deduced from the cosmic microwave background modelling. FLRW model is spatially homogeneous. The matter density is equal in every position in space. One of the attempts to understand the above issues is a consideration of inhomogeneous cosmological models. The work done in Department of Relativistic Astrophysics and Cosmology gives the inhomogeneous cosmological model constructed within the second-order perturbation theory.

The perturbation theory deals with a small corrections to the background solution, which in our case is the FLRW model. In cosmology, the linear perturbation theory is well understood. In this framework, one should leave in the equations the terms proportional to the perturbation but neglect the quadratic and higher-order terms. Unfortunately, the current-day inhomogeneities cannot be modelled satisfactorily by the linear perturbation theory. The correction under consideration should be small enough to fulfil the criteria of the linear perturbation theory, while the difference between the density at a galaxy cluster centre and the mean density of the universe is too big. The higher-order perturbations are then justified.

We construct every cosmological solution within the framework of General Relativity, which is our best theory of gravitation. Its fundamental components are the Einstein equations — the set of complicated non-linear partial differential equations. Because a complexity of these equations, it is not easy to find a solution to the cosmological perturbation theory in the order higher than linear. In the presented article, the author simplifies Einstein equations by imposing specific symmetry conditions. In this way, the second-order solution to the cosmological perturbation theory is constructed.

This work generalizes the earlier results [Maj 2019], where the background spacetime was spatially flat. The spatial flatness of the background is a common simplification in the perturbation theory. The actual paper is not restricted to the spatially flat background. Any FLRW model filled with dust matter and possibly non-zero cosmological constant can be the taken as the background spacetime, even for a non-zero spatial curvature. The most general FLRW model perturbed up to the second order in the presented way gives us a possibility to answer interesting question: how the local structures, modelled by these perturbations, affect cosmological observables?




Szymon Sikora

Astronomical Observatory
Jagiellonian University
S.Sikora [@]

Original publication

S. Sikora, Second-order dust perturbations of the non-flat FLRW model with the positive cosmological constant, Classical and Quantum Gravity 40, 025002, (2023).

The research was conducted at the Department of Relativistic Astrophysics and Cosmology of the Jagiellonian University’s Astronomical Observatory (OA UJ).

Gravitational lensing of blazar S5 0716+714

Blazars are a class of active galaxies whose jets (relativistic streams) are directed toward an Earth-based observer. They are characterized by high variability in luminosity, observed in various ranges of electromagnetic radiation (from radio to gamma rays) and on different time scales - from minutes to tens or even hundreds of years.

Gravitational lensing of blazar S5 0716+714

Illustration: The variation in object luminosity across three different observations. Credit: The Authors.


One of the well-known blazars is 0716+714. It has earned its fame among astronomers for its particularly high variability in the optical range of radiation, on time scales of the order of single nights. This makes it a great target for observing campaigns such as WEBT (Whole Earth Blazar Telescope), continuously monitoring the variability of blazars for several days in a row. Two WEBT datasets from 2003 and 2014 showed the characteristic pattern of blazar luminosity variation observed in each of the telescope's filters (Fig.1, green and red).

The object was also recently observed by the Transiting Exoplanet Survey Satellite (TESS), which provided us with its light curves for a total of 120 days. We have found several other manifestations of variability in it, with the same characteristics (Fig.1, black). This type of variability pattern is hard to relate to the currently known processes occurring in blazars.

We postulate in our work that the variability of the blazar, as observed several times, may be related to gravitational lensing of segments of its jet by a binary system located in the blazar's home galaxy. Gravitational lensing is the physical phenomenon of bending the course of light rays near massive objects by curving the space-time itself. This leads to their focusing, which, depending on the mass, configuration and mutual distances of the lens, observer and source, manifests itself in our observations in different ways. The lens, which is a binary system that passes in front of the line connecting the distant source to the observer, can trigger luminosity changes of precisely the type we observe in the luminosity light curve of S5 0716+714 blazar.

The results of our modelling show that this type of variation can be explained by the lens being a binary system consisting of a central main mass and an approximately 100 times less massive companion. The mass of the binary system depends on its position in the blazar's parent galaxy. Using the constraints imposed on the system due to the duration of events and the period between the first and last observations of the phenomenon, we obtain a mass of the lens that ranges from 104 to 106 solar masses. This could mean that the lens is a black hole of intermediate mass, that is, larger than the stellar-mass black holes formed as the final stage in the evolution of the most massive stars, and much smaller than the supermassive black holes observed in the centers of galaxies.




Dominika Król

Astronomical Observatory
Jagiellonian University
D.Krol [@]

Original publication

D. Król et al., Possible Gravitational Microlensing Events in the Optical Lightcurve of Active Galaxy S5 0716+714, The Astrophysical Journal 943 (2023).

The research was conducted at the Department of High Energy Astrophysics of the Jagiellonian University’s Astronomical Observatory (OA UJ).

Periodic Oscillations in LASCO Coronal Mass Ejection Speeds: Space Seismology

Coronal mass ejections (CMEs) are energetic expulsions of organized magnetized plasma from the Sun. They are intensively studied because of their significant impact on Earth. The first CME was discovered in 1971 using the seventh Orbiting Solar Observatory (OSO-7). The longest series of observations have been carried out by the Large Angle and Spectrometric Coronagraph (LASCO) on board the Solar and Heliospheric Observatory (SOHO) mission.

Sun cycle

Illustration 1: Examples of fitted curves (dashed lines) to speed–distance/time profiles for CME on February 2011. The upper panel displays speed–distance profiles and the lower panel speed–time profile. We observe a clear speed oscillation with an amplitude of about 250 m s1. Credit: The Authors.


Now we know that the magnetic field in CMEs and in the sheath ahead of shock-driving CMEs, especially the strength of the magnetic field component oriented in the direction opposite to that of Earth’s horizontal magnetic field (Bz), is crucial in causing geomagnetic storms along with the speed (V) with which the CME hits Earth’s magnetosphere. Unfortunately, routine measurements of magnetic fields, due to high temperature, in the outer solar atmosphere, where CMEs originate, are not available. To determine the magnetic field in the solar corona, we must apply indirect methods. It seems that one such method may be based on oscillations in the speed of CME. CMEs have an organized flux-rope magnetic structure. Therefore, the reconnection of the magnetic field during ejection may excite periodic speed oscillations. Periodic speed and intensity oscillations are well established in the case of quiescent prominences, which in many cases are progenitors of CMEs. These oscillations have a range of periodicity from a few minutes up to the order of an hour. We considered all the CMEs with at least 11 height–time measurements included in the CDAW database in the period of 1996–2019. It allowed us to evaluate speed–time profiles with enough accuracy to analyze oscillations in the speed of CMEs. Of the considered CMEs, 22% revealed periodic speed fluctuations during their expansion in the interplanetary medium. This means that speed oscillation is a frequent phenomenon associated with CME propagation.

We have obtained the following new important results. We demonstrated that the average values of the basic attributes of oscillations (amplitude and period) are significantly correlated with cycles of solar activity. In the maxima of solar activity, on average, pulsation periods are ≈240 minutes and in minima as much as ≈300 minutes. This means that periods of oscillations depend on the physical properties of ejections. It is assumed that the flux-rope structure inside CMEs can be approximated as a stretched elastic string of non-uniform density. Using this approximation, we estimated that, on average, the CME internal magnetic field varies from 18 up to 25 mG between the minimum and maximum of solar activity.


Slonce aktywne

Illustration 2: Temporal evolution of average annual values of solar spot numbers (panel a), speed of CMEs revealing oscillations (panel b), periods of oscillations (panel c), and their amplitudes (panel d). Error bars show the standard deviation for a given sample. Dashed lines in panels (c) and (d) represent the results obtained with a reduced cadence of height–time points to match the SC 23 cadence. Credit: The Authors.



The obtained results show that a detailed analysis of speed oscillations can be a very efficient tool for studying not only the physical properties of the ejections themselves but also the condition of the interplanetary medium in which they expand.




Grzegorz Michałek

Astronomical Observatory
Jagiellonian University
G.Michalek [@]

Original publication

Grzegorz Michalek et al., Periodic Oscillations in LASCO Coronal Mass Ejection Speeds: Space Seismology, The Astrophysical Journal Letters (marzec 2022)

The research was conducted at the Department of High Energy Astrophysics of the Jagiellonian University’s Astronomical Observatory (OA UJ).

13 years of monitoring the optical variability of quasars at OAUJ

Active galaxies are some of the most energetic objects in the Universe. Processes of accretion of matter onto supermassive black holes occur in their active nuclei (AGNs), constituting a source of radiation over a wide range of the electromagnetic spectrum. They are also objects exhibiting radiation variability. Studying the nature of this observed variability we can analyse the physical processes responsible for changes in the brightness of the active nucleus.

Monitoring quasars

Illustration 1: Left: Optical image from the Pan-STARRS survey, centred on one of the observed quasar [HB89] 1721+343. The optical image has been overlaid with radio contours from the NVSS (black colour) and FIRST (red) surveys. Right: Thirteen-year light curve of [HB89] 1721+343 quasar. Credit: The Authors.


The physical processes behind the variability in the optical range (on time scales from days to decades) are the subject of numerous studies. Theoretical considerations suggest that several mechanisms can explain both the time scales and the amplitudes of the brightness variations observed in AGNs. These may include instabilities in the accretion disk, processes associated with the evolution of massive stars that explode as supernovae, or variabilities caused by the gravitational microlensing. Each of these mechanisms results in a different character of variability, possible to study using statistical methods.

In 2009, systematic monitoring of the optical variability of the eight quasars has started in our Observatory. The selected objects have extended radio structures, with five of which are classified as so-called giant radio quasars, with sizes exceeding 0.7 Mpc. Initially, observations were performed using a fifty centimetre Cassegrain telescope located at the Astronomical Observatory of the Jagiellonian University. However, after some time, the telescope located at Suhora Observatory (UP) has also been used for the observations, as well as the telescopes belonging to the network of robotic telescopes – SKYNET. Thanks to the many people involved in the observations, it has been possible to obtain the light curves of the quasars, covering 13-year period of their variability. These are unique data not only because of the length of the observations period in question, but also due to the frequent sampling – the observations of each quasar were made on average 3 times a month.

The obtained light curves were used to investigate the characteristic time scales of the variation and the physical process that would explain the observed nature of the variation. For this purpose, two statistical methods was used: the so-called structure function (FS) analysis and power spectral density analysis (PSD). Based on the obtained FS and PSD slopes, it can be estimated that the most likely process that could be responsible for the variability of the observed quasars is the instabilities of the accretion disk, and the nature of this variability can be described as red noise or a damped random-walk process. Our analyses have also shown that the characteristic time scales of the variability of this type of objects are much longer than their monitoring time so far. They can span up to several hundred or even millions years, which is not achievable using conventional single-object observation methods.

An interesting though not fully understood obtained result is a strong anti-correlation between the size of radio structures and the slope of the PSD function (correlation coefficient of 0.86). The anti-correlation may indicate some link between the size of the radio source and the nature of its variability, but it is now based on data for eight objects only, making further study of a larger sample of this kind of objects necessary to confirm it.

Po lewej widzimy korelację pomiędzy jasnością mierzoną w zakresie promieniowania X a jasnością w zakresie optycznym dla kwazarów, bez poprawki na bias Malmquista. Po prawej widzimy tę samą korelację, ale poprawioną na opisany efekt. Po lewej widzimy wyraźną zależność korelacji od przesunięcia ku czerwieni, podczas gdy po prawej takiej zależności nie ma. Źródło: Oryginalna publikacja.


Illustration 2: Structure function for [HB89] 1721+343. The plot shows the value of the coefficient of the FS slope (α) and the inflection point (τ) corresponding to the characteristic time scale of the variability. Right: PSD function for [HB89] 1721+343 with the value of PSD slope (β) given. Credit: The Authors.


Rysunek 3. Wartości stałej Hubble’a H0 uzyskane dla różnych próbek i metod. Szare pionowe linie reprezentują wartości H0 otrzymane z użyciem mikrofalowego promieniowania tła (po lewej) i supernowych (po prawej). Po dodaniu kwazarów do analizy obejmujących supernowe można zauważyć, że otrzymane wartości H0 znajdują się pomiędzy wcześniej opisanymi wartościami. Źródło: Oryginalna publikacja.

 Illustration 3: The anti-correlation between the slope of the PSD function (β) and the size of the radio structure of quasars. Credit: The Authors.



Agnieszka Kuźmicz

Astronomical Observatory
Jagiellonian University
A.Kuzmicz [@]

Original publication

A Kuźmicz et al., Optical Variability of Eight FRII-type Quasars with 13 yr Photometric Light Curves, ApJS 263 16 (2022).

The research was conducted at the Department of Stellar and Extragalactic Astronomy and Department of Radioastronomy and Space Physics of the Jagiellonian University’s Astronomical Observatory (OA UJ). This research was supported by the Polish National Science Centre grants UMO-2018/29/B/ST9/01793 and UMO-2018/29/B/ST9/02298. The quasar light-curve simulations were performed using the PLGrid Infrastructure.

Crisis in Cosmology: statistics on the rescue

International team of scientists from Italy, Poland, and Japan has released a groundbreaking article with a solution to one of the biggest challenges of modern observational cosmology. A new statistical method was applied in order to remove the impact of selection biases on cosmological computations while avoiding circular reasoning. The article has been accepted for publication in The Astrophysical Journal Supplement Series.

Rysunek 1. W centrum zdjęcia widzimy Kwazar poddany soczewkowaniu grawitacyjnemu przez galaktykę znajdującą się przed nim. Cztery równo ułożone punkty są obrazami tego samego Kwazara, powstałe poprzez grawitacyjne zakrzywianie promieni światła przez soczewkującą galaktykę. Źródło: ESA/Hubble, NASA, Suyu et al. (Lensed quasar and its surroundings | ESA/Hubble (

Figure 1. In the center of the image, there is a quasar lensed by the galaxy. Four evenly distributed dots are images of the same quasar, as an effect of gravitational lensing of beams of light by the galaxy in front of the quasar. Credit: ESA/Hubble, NASA, Suyu et al.


Modern cosmology has many open questions: how to solve the discrepancy between the measurements of the Hubble constant? What is the curvature of space-time? What is the nature of dark energy? And many, many more. Astronomers are trying to shed new light on these issues by observing the Universe farther and farther away from the Earth. This is driven by the fact that predictions from different cosmological models have minor differences at a low distance. Still, those differences begin to be significant for objects very far away from us.

In order to probe the high-distance Universe, scientists have to use the most luminous objects that ever existed, quasars and Gamma-Ray Bursts. Quasars are powered by the accretion of gas onto the supermassive black holes, residing in the centers of galaxies. Gamma-Ray Bursts originate when a massive star explodes, or when two neutron stars, or a neutron star and a black hole merge. Since the discovery of these objects, many correlations between their physical quantities have been studied.

These correlations allow us to determine the distance of these objects from us. This is crucial for cosmological analyses, since we have the theoretical formula for the distance in a given cosmological model, which includes the speed of the expansion of the given point in the Universe. This given speed can be measured spectroscopically, thus we can compute the theoretical distance of a particular object for a given cosmological model. The comparison of the theoretical distance and the observed one allows us to test how well the given cosmological model fits the data.

But looking at high distances has shortcomings: the farther away we look, the less luminous events we see. This phenomenon is called the Malmquist bias effect. This can induce an artificial correlation between the observed parameters, so how do we know if the given correlation is a physical feature of the objects, or just induced by a selection effect? This can be tested via a method proposed by Efron & Petrosian (1992) and already successfully developed for the Risaliti-Lusso relation of the quasars by Dainotti et al. 2022 (article by the same authors as the mainly discussed work). This technique eliminates the correlation between the studied parameters and the observational spectroscopic parameter determining the distance (the redshift).

This method creates yet another problem. Many correlations involve luminosity, and in order to calculate the luminosity, one needs to fix the parameters of the cosmological model. Then, with those fixed values, one determines the correction for the selection bias effect. Consequently, this correction has been applied within the assumed cosmological model. Thus, if we would like to use these corrected values to test some cosmological model, our results will be again biased by the model assumed a priori. This represents the so-called circular reasoning problem and it cannot be applied to scientific approaches.

The new article completely overcomes this issue, by creating a new, more general treatment than the one of Efron & Petrosian (1992). This treatment was first discussed by Dainotti et al. (2022), and later has been fully implemented by Lenart & Bargiacchi et. al (2023). Lenart & Bargiacchi have proposed a method which treats the correction for selection bias as a function of the cosmological parameters, allowing to obtain a general formula of correction, valid for any value of cosmological parameters, which can be applied without assuming any values a priori.

The application of this method to the so-called Risaliti-Lusso correlation linking two luminosities for quasars allowed for the first time in the literature to obtain a 2 sigma value of ΩM, the parameter denoting the current amount of matter in the Universe. Moreover, the authors applied the following analysis to study the impact of high-distance measurements on the so-called H0 tension (the incompatibility of the two measurements of the Hubble constant: the one involving supernovae type Ia and the one obtained from the Microwave Cosmic Radiation). Surprisingly, quasars seem to prefer the value of the Hubble constant in between the values obtained for supernovae type Ia and Microwave Cosmic Radiation. This suggests that this discrepancy might be driven by some still unknown physics, and not by the statistical effects. Although in order to confirm such a conclusion, the new correction has yet to be applied also to other samples like supernovae type Ia. The overall results enhance a further significant improvement in cosmological measurements and an extension of our knowledge about the Universe at high distances.

Po lewej widzimy korelację pomiędzy jasnością mierzoną w zakresie promieniowania X a jasnością w zakresie optycznym dla kwazarów, bez poprawki na bias Malmquista. Po prawej widzimy tę samą korelację, ale poprawioną na opisany efekt. Po lewej widzimy wyraźną zależność korelacji od przesunięcia ku czerwieni, podczas gdy po prawej takiej zależności nie ma. Źródło: Oryginalna publikacja.


Figure 2. On the left, uncorrected correlation between luminosity measured in an X-ray band and visible light, while on the right, the same correlation, but corrected for evolution and selection bias with the described method. On the left, we can see a clear evolution of the correlation with redshift, while on the right no such evolution can be spotted. Credit: Original publication.


Rysunek 3. Wartości stałej Hubble’a H0 uzyskane dla różnych próbek i metod. Szare pionowe linie reprezentują wartości H0 otrzymane z użyciem mikrofalowego promieniowania tła (po lewej) i supernowych (po prawej). Po dodaniu kwazarów do analizy obejmujących supernowe można zauważyć, że otrzymane wartości H0 znajdują się pomiędzy wcześniej opisanymi wartościami. Źródło: Oryginalna publikacja.


Figure 3. The values of the H0 obtained for different samples and methods. Grey vertical lines represent the values of H0 computed with the Cosmic Microwave Background radiation data (on the left) and supernovae Ia data (on the right). After the addition of quasars data to the supernovae Ia, one can spot that results fall perfectly between the two before mentioned values. Credit: Original publication.



Aleksander Lenart

Astronomical Observatory
Jagiellonian University
Aleksander.Lenart [@]

Original publication

Aleksander Łukasz Lenart, Giada Bargiacchi, Maria Giovanna Dainotti, Shigehiro Nagataki, Salvatore Capozziello, A bias-free cosmological analysis with quasars alleviating H0 tensionarXiv:2211.10785 [astro-ph.CO] (2022).

The publication under discussion was made possible, amongst others, thanks to financial support from the Astronomical Observatory of Jagiellonian University in Cracow and the Program Council for Mathematical and Natural Sciences Studies at Jagiellonian University. We also acknowledge the National Astronomical Observatory of Japan and RIKEN for their support in realising the work and partially funding the publication.